Theorems · Theorem · linear algebra

LinearMap.ker_le_range_iff

∀ {R : Type u_1} {M : Type u_2} {R₂ : Type u_3} {M₂ : Type u_4} {R₃ : Type u_5} {M₃ : Type u_6} [inst : Ring R]
  [inst_1 : Ring R₂] [inst_2 : Ring R₃] [inst_3 : AddCommMonoid M] [inst_4 : AddCommGroup M₂]
  [inst_5 : AddCommMonoid M₃] [inst_6 : Module R M] [inst_7 : Module R₂ M₂] [inst_8 : Module R₃ M₃] {τ₁₂ : R →+* R₂}
  {τ₂₃ : R₂ →+* R₃} [inst_9 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {g : M₂ →ₛₗ[τ₂₃] M₃},
  g.ker ≤ f.range ↔ f.range.mkQ ∘ₗ g.ker.subtype = 0

Defined in

Mathlib.LinearAlgebra.Quotient.Basic

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingRingRingAddCommMonoidAddCommGroupAddCommMonoidModuleModuleModuleRingHomSurjective